Let {Z n , n = 0, 1, 2, . . .} be a supercritical branching process, {N t , t ≥ 0} be a Poisson process independent of {Z n , n = 0, 1, 2, . . .}, then {Z N t , t ≥ 0} is a supercritical Poisson random indexed branching process. We show a law of large numbers, central limit theorem, and large and moderate deviation principles for log Z N t .

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We study the minimum Skorohod distance estimation and minimum -norm estimation of the drift parameter θ of a stochastic differential equation , , where is a fractional Lévy process, . We obtain their consistency and limit distribution for fixed T, when . Moreover, we also study the asymptotic laws of their limit distributions for .